Telicity: Difference between revisions

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'''Telicity''' is a property of both [[EDO]]s and [[commas]] and how they relate to each other. Specifically, for EDOs, it is the quality or state of being able, by means of tempering one or more commas smaller than half a step, to successfully stack a number of instances of a given prime's [[patent interval]] to connect with an interval belonging to a chain created by a lower prime's patent interval – designated as the '''telos''' – without either accumulating 50% relative error or more at any point in the process on the part of either prime's patent interval chain, or, creating as mismatch in results between the direct mapping and the more complicated traditional mapping for any interval along the chain. For commas, telicity is the quality of being able to join two distinct prime interval chains in the aforementioned manner by being tempered.

Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two prime chains in question~~, and "M" is the monzo of "r"~~, the following equation must hold true along both prime chains up to and including the point of connection:

Stated more mathematically, where ''N'' is the number of steps in a given EDO, ''r'' is the ratio of an interval in one of the two prime chains in question, the following equation must hold true along both prime chains up to and including the point of connection:

{~~N, round~~(log2(3)*N)~~, round~~(~~log2(5~~)*N), round(~~log2~~(7)*N)~~, round~~(~~log2(11~~)*N), .~~..}.{M} = round(log2(r)*N)~~

<math>\text{val} (N) \cdot \text{monzo} (r) = \text{round} (N \log_2 (r))</math>

where val (''N'') denotes the [[patent val]] of ''N''-EDO.

Commas and EDOs that demonstrate this property are referred to as as being '''telic'''. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.

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 '''Telicity''' is a property of both [[EDO]]s and [[commas]] and how they relate to each other.  Specifically, for EDOs, it is the quality or state of being able, by means of tempering one or more commas smaller than half a step, to successfully stack a number of instances of a given prime's [[patent interval]] to connect with an interval belonging to a chain created by a lower prime's patent interval – designated as the '''telos''' – without either accumulating 50% relative error or more at any point in the process on the part of either prime's patent interval chain, or, creating as mismatch in results between the direct mapping and the more complicated traditional mapping for any interval along the chain.  For commas, telicity is the quality of being able to join two distinct prime interval chains in the aforementioned manner by being tempered.
-Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two prime chains in question, and "M" is the monzo of "r", the following equation must hold true along both prime chains up to and including the point of connection:
+Stated more mathematically, where ''N'' is the number of steps in a given EDO, ''r'' is the ratio of an interval in one of the two prime chains in question, the following equation must hold true along both prime chains up to and including the point of connection:
- {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N)
+<math>\text{val} (N) \cdot \text{monzo} (r) = \text{round} (N \log_2 (r))</math>
+where val (''N'') denotes the [[patent val]] of ''N''-EDO.
 Commas and EDOs that demonstrate this property are referred to as as being '''telic'''.  When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.